An Evolutionary Strategy for. Decremental Multi-Objective Optimization Problems

Transcription

1 An Evolutionary Strategy or Decremental Multi-Objective Optimization Problems Sheng-Uei Guan *, Qian Chen and Wenting Mo School o Engineering and Design, Brunel University, UK Department o Electrical and Computer Engineering National University o Singapore Abstract In this paper, an evolutionary algorithm or multi-objective optimization problems in a dynamic environment is studied. In particular, we ocus on decremental multi-objective optimization problems, where some objectives may be deleted during evolution - or such a process we call it as objective decrement. It is shown that the Pareto-optimal set ater objective decrement is actually a subset o the Pareto-optimal set beore objective decrement. Based on this observation, the inheritance strategy is suggested. When objective decrement takes place, this strategy selects good chromosomes according to the decremented objective set rom the solutions ound beore objective decrement, and then continues to optimize them via evolution or the decremented objective set. The experimental results showed that this strategy can help MOGAs achieve better perormance than MOGAs without using the strategy, where the evolution is restarted when objective decrement occurs. More solutions with better quality are ound during the same time span. Keywords: multi-objective problems, multi-objective genetic algorithms, multi-objective optimization, vector optimization, non-stationary environment * Corresponding author

2 . Introduction Multi-objective optimization problems (MOP) require simultaneous optimization o several objectives which may compete against each other. Mathematically, an MOP with n variables and p objectives aims to ind x = x,..., x ) which minimizes the values o ( n p objectives =,..., ) within the easible input space. Since any maximization ( p objective can be turned into a minimization one, this paper considers minimization only. In our research, the ollowing deinitions are used throughout:. The easible input space I is the set o decision vectors (solutions) that satisy the constraints and bounds o the problem. The easible objective space O is the set o objective vectors (points) with respect to each point in I.. Let x, y I, y is said to be dominated by (or inerior to) x under, i i ( x) i ( y) i =,,..., p AND j ( x) < j ( y) j (,,..., p) 3. Let r I be a Pareto-optimal solution i there is no other solution x I that dominates r. The set o Pareto-optimal solutions is the Pareto-optimal set or the given problem. Generally, due to the presence o multiple objectives, the answer to an MOP consists o a set o Pareto-optimal solutions instead o only one optimal solution. Without urther inormation, one Pareto-optimal solution cannot be declared as better than another. So, optimization methods usually try to ind a number o Pareto-optimal solutions which are uniormly distributed in the Pareto-optimal set to provide the decision maker suicient insight into the problem to make the inal decision.

3 Because Genetic Algorithm (GA) maintains a population and can ind a diverse set o solutions in a single run, multi-objective GA (MOGA) has received a lot o interests and a number o approaches [9-] have been suggested. NSGA-II [9], SPEA [, ] and PAES [3] are well-received MOGAs. Basically, an MOGA is characterized by its itness assignment and diversity maintenance strategy. In itness assignment, most MOGAs all into two categories, Non-Pareto and Paretobased [7, 8]. Non-Pareto methods [4, 9, 0] directly use the objective values as individuals itness. Schaer s VEGA [9] is an example. In contrast, Pareto-Based methods [9-3, 8] measure each individual s itness according to its dominance property. NSGA-II, SPEA and PAES all belong to the latter category. Since Pareto-based methods respect better the dominance nature o MOPs, their perormance is reported to be better. Diversity maintenance strategy is another characteristic o MOGAs. It keeps the solutions more uniormly distributed, instead o gathering in a small region. Fitness sharing [], which reduces the itness o an individual i there are some other candidates nearby, is one o the most renowned techniques. More recently, some parameter-ree techniques were suggested. The techniques used in SPEA [, ] and NSGA-II [9] are two examples. Readers are encouraged to turn to the original papers or details. The MOGAs described above all deal with a static objective set, which does not vary with time. But in the real world, MOPs may arise rom more complex environments, where the objective set may vary with time. For example, new objectives may be added 3

4 or obsolete objectives may be deleted, some objectives may be changed, or even all the objectives may be changed, etc. Actually there have been a lot o interests in applying GA to time-varying singleobjective optimization problems. Many techniques have been proposed [-6]. For example, some researchers suggested the hyper-mutation strategy, which inserts new random members into the population periodically []. Other researchers suggested chromosomes to be selected according to a combined unction o the optimization objective value and the age o a chromosome, the younger ones are more likely to survive [5]. And some others tried to divide the population into multiple species, where the crossover between dierent species are restricted, thus diversity is preserved and the population is more responsive to changes [4, 6]. However, as ar as we know, none o these techniques considered multi-objective optimization. Dynamic MOPs are more complicated with various types o changes. Dierent types o changes may require dierent evolutionary algorithms. This paper studies one speciic type o changes, namely objective decrement, which means one or more objectives may be deleted rom the objective set during evolution. We call this type o problems as decremental multi-objective optimization problems. We aim at inding some multiobjective GA strategy that is more responsive to objective decrement. In real world, the decremental multi-objective optimization problems do make sense and can be applied to a lot o dynamic situations. As an example, an allocation o provide projects to each inal year student is conducted every year in ECE department o NUS. Initially, this job was 4

5 done by an MOGA to maximize the satisaction o the students as well as to obtain a uniorm distribution o students or all the projects. However, it s regarded that students satisaction plays a key role in their perormance. So the objective was changed to only maximize the students satisaction. Another example is the unit commitment and generation scheduling in power system. The initially used objective set is to minimize the cost, to maximize the saety and to minimize the pollution. But ater some time, the cost is not still a problem and then it may be deleted rom the objective set. Actually, some eort on the uncertainty o objective unctions was made by Jurgen Teich [8]. However, he just proposed some probabilistic dominance criterion or the situation where the objective values vary within intervals, rather than the objective unction set changes. We analyze the relationship between the Pareto-optimal sets beore and ater objective decrement. It is shown that in general, the Pareto-optimal set beore decrement is actually a superset o the Pareto-optimal set ater decrement. The inheritance strategy is proposed to utilize this act to achieve better perormance. The experimental results showed that the suggested strategy, which selects and reuses chromosomes rom the solutions evolved or the original objective set when objective decrement occurs, can help ind more solutions with better quality within the same time span. The results o this research can ind applications in situations where an MOP problem is pre-deined and solutions have been derived, yet one or more objectives are dropped (or replaced) on the run. Solutions then have to be derived timely or the changed MOP. For 5

6 objectives that are replaced, we can treat the problem as an MOP problem with objective decrement ollowed by objective increment immediately. In the ollowing, Section analyzes the relationship between the Pareto-optimal sets beore and ater objective increment. Three theorems about the relationship are proved. Based on Section, Section 3 deduces the relationship between the Pareto-optimal sets beore and ater objective decrement. Section 4 describes the inheritance strategy to cope with objective decrement. Section 5 presents the experimental results and analysis. Section 6 discusses the applicability o inheritance strategy. Section 7 concludes this paper.. Objective Increment and Its Eect Since objective decrement is just a reverse operation o objective increment and a orward process is always easier to understand than a reverse one. We will irstly analyze the eect o objective increment in this section. And based on the conclusion, the eect o objective decrement will be inerred in the next section.. Deinitions Assume the initial objective set is F =,..., ). Later v new objectives are added, ( u and the objective set becomes F + = (,... u, u+,..., u+ v ). We call this process as objective increment. F + is the incremented objective set, and F = ( u +,..., u + v ) represents the objectives added. We give the ollowing deinitions to acilitate discussion: 6

7 . I two points in I give equal output in every objective, we say these two points are phenotypically equal to each other under the speciied objective set. Similarly, i two points give dierent output in one or more objectives, they are phenotypically distinct under the speciied objective set.. A Pareto-optimal solution x is a unique Pareto-optimal solution (abbreviated as unique P-solution) i there is no other solution phenotypically equal to x within I. 3. A Pareto-optimal solution x is a non-unique Pareto-optimal point (abbreviated as non-unique P-solution) i there is one or more solutions phenotypically equal to x within I. 4. A non-unique Pareto-optimal solution together with all the solutions phenotypically equal to it within I constitutes a non-unique Pareto-optimal group (abbreviated as non-unique P-group). 5. A Pareto-optimal output p is a point (vector) in O which corresponds to a Pareto-optimal solution s in I, namely p = ( ( s), ( s),..., p ( s)). Obviously, a unique P-solution corresponds to one Pareto-optimal output, while all the solutions in a non-unique P-group correspond to one Pareto-optimal output. In addition, the ollowing notations are used in this section: P is the Pareto ront beore objective increment. P + is the Pareto ront outputs ater objective increment. 7

8 ' P + is P + with every member s elements corresponding to F being truncated. For example, assume F = (, ), F + = (,, 3, 4 ), F = ( 3, 4 ), and 3 4 πx = sin πx = cos = ( x 3) = x with the constraint: x (0,,,3,...,00 ) then P + ={(0,,9,0),(,0,4,),(0,,,4),(,0,0,9)}, and ' P + ={(0,),(,0)}. Please note that the replica members ater truncation should be deleted. S is the Pareto set beore objective increment. S + is the Pareto set ater objective increment.. Eect o Objective Increment on Pareto-optimal Points.. Eect on Unique P-solutions Theorem : I a point x is unique Pareto-optimal under the initial objective set F, x will remain unique Pareto-optimal under the incremented objective set F +. Proo: Apagoge is used. Assume the point x is not unique Pareto-optimal under F +. That is, there exists another point x ' in the easible input space under F +, which either:. dominates x. That is: i ( x') i ( x) i =,,..., u + v () 8

9 . or is phenotypically equal to x. That is: i ( x') = i ( x) i =,,..., u + v () From inequality () and equation (), we can deduce that i ( x') i ( x) i =,,..., u (3) Inequality (3) means that, under F, x is either dominated by x ' or phenotypically equal to x '. Namely, under F, x is either non-pareto-optimal or nonunique Pareto-optimal. This conclusion contradicts with the premise. Thus, Theorem is proved... Eect on Non-unique Solutions The outcome o a non-unique P-solution x, whose non-unique P-group is denoted as ater objective increment may be: W x,. x becomes non-pareto-optimal, i one or more points in Wx dominate x under F.. x remains non-unique Pareto-optimal, i no other point in W x dominates x and one or more points in W are phenotypically equal to x under 3. x becomes unique Pareto-optimal, i no other point in W x dominates x or remains phenotypically equal to x under F. F. Though the outcome o a non-unique P-solution ater objective increment is nondeterministic, the outcome o a non-unique P-group ollows some rule, as stated in the ollowing theorem: 9

10 Theorem : For a non-unique P-group W under the initial objective set F, at least one member solution will remain Pareto-optimal under the incremented objective set F +. Proo: Apagoge is used. Assume no solution in W is Pareto-optimal under F +. That is, under F + any solution in W would be dominated by one or more solutions not belonging to W. Now, assume one speciic solution in W, x, is dominated by a point x ' which does not belong to W. Then: i ( x') i ( x) i =,,..., u + v (4) so, under the original objective set F : i ( x') i ( x) i =,,..., u (5) Inequality (5) means that, x is either dominated by x ' or phenotypically equal to x ' under F. Namely, under F, either x is non-pareto-optimal, or there exists a solution not belonging to W but phenotypically equal to x. This conclusion either contradicts with the premise or the deinition o a non-unique P-group. Thus, Theorem is proved. It is possible that there are more than one solution in W which are not dominated by any other solution in W under F, thus they will all become Pareto-optimal ater objective increment. I these non-dominated solutions are phenotypically distinct under F, they will correspond to more than one Pareto-optimal output in O or F +..3 Eect o Objective Increment on Non-Pareto-optimal Solutions 0

11 Denote the set o solutions that dominate or are phenotypically equal to a non-paretooptimal solution x beore objective increment as D x. The outcome o x ater objective increment is decided by its dominance relationship to D under F : x. x becomes Pareto-optimal, i no point in D x dominates it under F.. x remains non-pareto-optimal, i one or more points in D x dominate x under F. So, objective increment may turn some non-pareto-optimal points into Pareto-optimal..4 Eect on Pareto Set and Pareto ront Solutions in I can be classiied into three categories: unique P-solutions, non-unique P- solutions, and non-pareto-optimal solutions. The discussion above shows that objective increment brings the ollowing changes to the optimality status o the solutions under these three categories: ) a unique P-solution beore objective increment corresponds to a Pareto-optimal output, and still corresponds to a Pareto-optimal output ater objective increment. ) all the points in a non-unique P-group beore objective increment correspond to one Pareto-optimal output, and correspond to one or more Pareto-optimal outputs ater objective increment. 3) a non-pareto-optimal solution beore objective increment does not correspond to any Pareto-optimal output, yet it may correspond to one Pareto-optimal output ater objective increment.

12 ) and ) show that each member point (vector) in P, ater appending the elements corresponding to theorem: F properly, becomes a member o P +. So, we have the ollowing Theorem 3: ' P P + However, it must be noted that although ' P P +, S S + does not necessarily hold. The relationship between S and S + is a bit more complex. Those non-unique P-solutions which lose their optimality ater objective increment belong to S but not to S +. Generally, in real-world MOPs, especially those with many objectives, it is unlikely that many solutions get phenotypically equal. Thereore, unique P-solutions are generally more than non-unique ones. So the majority o S are strong points and thus belong to S +. However, in some special cases the majority o S are non-unique P-solutions and thus may not belong to S +. ' Given that I is a inite set o solutions (thus S, S +, S +, P and P + are all inite sets), we can reach the ollowing conclusions ( denotes the size o set ): ) because ' P P + and ' P + P +, we have P ' P + P +. ) usually the majority o S belong to S +, and some non-pareto-optimal points may become Pareto-optimal ater objective increment, so generally S S +.

13 3) in some special cases, the non-unique solutions degraded as non-pareto-optimal are more than those non-pareto-optimal solutions upgraded as Pareto-optimal, so under such circumstances S. An example is given in the Appendix. S + 3. Objective Decrement and Its Eect Assume the initial objective set is G = g,..., g ). Later n ( n < m) objectives are ( m deleted rom G, and the objective set becomes G g,..., ). We call this process = ( g m n as objective decrement. G stands or the decremented objective set, and G = + ( g m n,..., g m ) represents the objectives deleted. Please note that during objective decrement, any objective in G may be deleted. However, since our discussion has nothing to do with the ordering o objectives in G, we assume the deleted objectives are the ones with higher indices or the ease o narration. Following the notations used in Section,: P is the set o Pareto ront ater objective decrement. P is the set o Pareto ront beore objective decrement. P ' is the set P with each member s elements corresponding to G being truncated. S is the set o Pareto-optimal points ater objective decrement. S is the set o Pareto-optimal points beore objective decrement. 3

14 The eect o objective decrement can be inerred rom the discussions in section.4, since it is the reverse operation o objective increment. I I is a inite set o points: ) P P ', and P P ' P. ) usually, S contains the majority o S, and S S. 3) in some cases, i the majority o S are non-unique solutions which are not Pareto-optimal under G, then S might not contain the majority o S. And i these solutions are more than the solutions in S but non-pareto-optimal under then S S. G, Current MOGAs, such as NSGA-II, PAES and SPEA, try to ind points uniormly distributed in the objective space instead o the input space. So, the ollowing discussion ocuses on the relationship between P and P, instead o that between S and S. In GA applications, search is usually done within a limited range and a pre-deined sampling resolution, thereore I is a inite set. So, i the evolution has ound the majority o P when objective decrement occurs, generally the majority o P has already been ound. Even i P is ound only partially, there still exist some members o P in the population. I search is simply restarted, then the optima ound so ar will be discarded, which results in waste o computation eort. Hopeully, better perormance is available i we utilize those optima ound beore objective decrement. 4

15 Nevertheless, because the number o solutions in P may be much greater than the user needs, most MOGAs aim to ind a number o solutions uniormly distributed in P, instead o the whole P. In such cases, we can still expect that, in the solution set ound beore objective decrement, there exists some well-perorming solutions under use o them can make the algorithm more responsive to objective decrement. G, good 4. Inheritance Strategy The term inheritance is used in this paper to describe the process o making use o the population evolved beore objective decrement in the evolutions ater objective decrement, which can be illustrated by Figure. The analysis in Section and Section 3 shows the rationale or inheritance. In this section we suggest the inheritance strategy to be used in three MOGAs, namely PAES [3], SPEA [,] and NSGA-II [9]. For each algorithm, a brie review about the algorithm is ollowed by the description o inheritance strategy it uses. Eo Pareto Set Pareto optimal under Od? Y Inherited Ed N Discarded Figure Procedure o inheritance strategy In Figure, E o and E d respectively stand or evolution beore and ater objective decrement, and O d stands or the objective set ater objective decrement. 5

16 4. Inherit Strategy or PAES PAES is an MOGA using one-parent, one-ospring evolution strategy. The nondominated solutions ound so ar are stored in the so-called archive. When the archive is not ull, a new non-dominated solution will be accepted by the archive. When the archive is ull and yet a new non-dominated solution is ound, i the new solution resides in a least crowded region, it will be accepted and a copy is added to the archive, at the same time a solution in the archive which resides in the most crowded region is deleted. With PAES, our inheritance strategy works as ollows. When objective decrement happens, the solutions in the archive are compared in pairs under the decremented objective set and only those non-dominated solutions survive. The population is kept and then the evolution goes on under the decremented objective set, based on the updated archive. 4. Inheritance Strategy or SPEA In SPEA, an external population is maintained to store the non-dominated solutions discovered so ar. During each generation, the external population and the current population orm a combined population. All non-dominated solutions in the combined population are assigned a itness based on the number o solutions they dominate and dominated solutions are assigned a itness based on the itness o their dominating solutions. A clustering technique is used to ensure diversity in the external population. 6

17 With SPEA, our inheritance strategy works as ollows. When objective decrement occurs, the solutions in the external population are compared in pairs under the decremented objective set and only those non-dominated solutions will survive. The population is kept and then the evolution goes on under the decremented objective set, based on the updated external population. 4.3 Inheritance Strategy or NSGA-II In NSGA-II, in every generation, crossover and mutation are perormed to generate as many ospring as the parent population. Then the whole population is sorted based on non-domination and each solution is assigned a itness value equal to its non-domination level. The solutions belonging to a higher level are regarded as itter. I it is necessary to select solutions at the same level, the solutions will be compared based on their crowding distance. The itter hal o the population will survive. Since there is no speciic mechanism as the external population in SPEA or the archive in PAES to store the non-dominated solutions, or NSGA-II, our inheritance strategy is slightly dierent. When objective decrement occurs, the whole population is kept and the evolution goes on under the decremented objective set. 5. Experiments and Results 5. Perormance Evaluation Metrics Indicated by Zitzler [6], multi-objective optimization is quite dierent rom singleobjective optimization in that there is more than one goal: 7

18 ) Convergence to the Pareto-optimal set. ) Maximized extent o the non-dominated ront obtained. 3) A good (in most cases uniorm) distribution o the solutions ound. So, the perormance evaluation o multi-objective optimization is a non-trivial task. A lot o metrics have been proposed [3-7][9]. In this paper, the ollowing metrics are used, corresponding to the goals mentioned above: ). ϒ and σ ϒ are metrics describing the solutions convergence degree. To compute them, ind a set o true Pareto-optimal points uniormly spaced in the objective space irst. Then or each solution, we compute its minimum Euclidean distance to the true Pareto-optimal points. The average o these distances is ϒ, and the variance o the distances is σ ϒ. ϒ indicates the closeness o the solutions to the real Pareto-ront, and σ ϒ indicates how uniormly they approach the ront. ). The coverage o the solutions is described by the metric η, according to the volume-based scaling-independent S metric and D metric proposed by Zitzler [9] with some slight modiication, we deine: A Pareto ront ound by MOGA algorithms; B True Pareto ront ound by a brutal-orce method; ( ) V : = S B, hypervolume o the objevtive space dominated by the true Pareto ront; ( ) ( ) ( ) α : = D A, B = S A + B S B, hypervolume o the objective space dominated by the ound Pareto ront but not by the true Pareto ront; 8

19 ( ) ( ) ( ) β : = D B, A = S A + B S A, hypervolume o the objective space dominated by the true Pareto ont but not by the ound one; Here, V is set as the reerence volume and the comprehensive coverage metric η = α V + β V The aim is to measure the correctly covered objective space by the MOGA algorithms. I η is close to or larger than 0, the solutions can be regarded as just covering the majority o the Pareto ront. 3). σ d measures how uniorm the solutions spread. To compute σ d, or every solution, ind out its minimum normal Euclidean distance (denoted as N E. The deinition is given below. It is designed in such a way to avoid bias among objectives whose extents may be quite dierent) to the other solutions. The variance o these distances is σ d. The smaller σ d is, the better they are distributed. N E ( a) ( b) ( a, b) = ( ) n k k max min k = tk tk a and b are points in Pareto ront, ( ) k i is the objective value o the ound Pareto ront in the kth objective, and t max t min is the extent o true Pareto ront in k the kth objective. n is the number o objectives. 4). Besides the above metrics, another simple metric is also used, which is the number o solutions ound, L. More solutions give the decision maker more choices and a better inal decision is more likely. k 9

20 For MOP in a dynamic environment, it is important to keep up with the changes to ind out solutions beore the next change. Thereore the perormance comparison is timebased. The evolution time beore objective decrement is denoted as t o, and that or the decremented objective set is denoted as t d. 5. Experiment Scheme Overview To show the advantage o inheritance strategy, PAES, SPEA and NSGA-II with the inheritance strategy will be compared with each algorithm without the inheritance strategy, namely the search process restarts when the objective set changes. NSGA-II includes two encoding schemes, NSGA-II in real coding (shortened as NSGA-II(R)) and NSGA-II in binary coding (shortened as NSGA-II(B)). All the results are the average o 0 runs. In each run a dierent random sequence is used. For SPEA and PAES, the initial seeds are the ten integers rom to 0. The NSGA-II program needs decimal initial seeds between 0 and, so ten uniormly spaced decimals, 0, , are used. The programs or these algorithms were downloaded rom their developers websites. [PAES: iridia.ulb.ac.be/~jknowles/multi/paes.htm, SPEA: NSGA-II: All the experiments were done on a Pentium IV.0 GHz PC. The ollowing parameters are set according to their original papers and kept the same in all the experiments: 0

21 Mutation rate or each decision variable is / n and or each bit is /l ( n is the number o decision variables, l : the length o the chromosomes). For SPEA, the ratio o population size to the external population is 4:. For PAES, the depth value is equal to 4. For NSGA-II, the crossover probability is 0.9, and the distribution indices or crossover and mutation are η = 0 and η = 0. c m 5.3 Experimental Results 5.3. Experiments on Problem G = (,, 3 ) and G = (, ). This problem is adapted rom the benchmark problem ZDT and ZDT [6]. (ZDT is, ) and ZDT is, ).). ( ( 3 3 = x = g( x)[ = g( x)[ ( x g( x) = + 9( n i= x / g( x)] / g( x)) ] x i ) /( n ) x [0,] n = 30, i n i Each variable is encoded in 30 bits. 000 uniormly spaced Pareto-optimal solutions are ound or the computation o ϒ and σ ϒ. The population size o NSGA-II is set at 00, and the size o archive in PAES and the external population in SPEA is also set at 00. The ollowing notations are used in the ollowing tables: t o : evolution time beore objective decrement

22 t d : evolution time ater objective decrement L: number o solutions ound ϒ: average distance o solutions to Pareto ront σ ϒ : variance o distances rom solutions to Pareto ront η: solution coverage extent index σ d : solution distribution uniormity index restart: start an evolution under the objective set ater objective decrement rom a randomly initialized population. Table Perormance o SPEA restarting/inheritance on objective decrement (problem ) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A inherit e e-6 restart N/A inherit e e e e-6 Table Perormance o PAES restarting/inheritance on objective decrement (problem ) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A inherit restart N/A inherit e-5

23 Table 3 Perormance o NSGA-II(R) restarting/inheritance on objective decrement (problem ) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A inherit e e e e e-6 restart N/A inherit e e e e e e-6 Table 4 Perormance o NSGA-II(B) restarting/inheritance on objective decrement (problem ) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A inherit restart N/A inherit e-5 Discussions on the results are given in section Experiments on Problem G = (,, 3 ) and G = (, ). This problem is adapted rom the benchmark problem FON [6]. G is problem FON. 3

24 3 = exp( = exp( = n i= x i n i= n i= ( xi ( x + i ) 3 ) 3 ) ) where x [ 4,4], n = 3 i Each variable is encoded in 30 bits. 000 Pareto-optimal solutions are ound or the evaluation o ϒ and σ ϒ. The population size o NSGA-II is set at 00, and the size o archive in PAES and the external population in SPEA is also set at 00. As to the legends o notations used in the ollowing tables, please reer to Section Table 5 Perormance o SPEA restarting/inheritance on objective decrement (problem ) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A e-5 inherit e e e e e e-6 restart N/A e-5 inherit e e e e e e-6 Table 6 Perormance o PAES restarting/inheritance on objective decrement (problem ) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A e e-5 Inherit e e e e e e-6 restart N/A e e-6 Inherit e e e e e e-6 4

25 Table 7 Perormance o NSGA-II (R) restarting/inheritance on objective decrement (problem ) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A e e-6 Inherit e e e e e e-6 restart N/A e e-6 Inherit e e e e e e-6 Table 8 Perormance o NSGA-II (B) restarting/inheritance on objective decrement (problem ) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A e-7 Inherit e e e-6 restart N/A e e-6 Inherit e e e e e e-6 Discussions on the results are given in section Experiments on Problem 3 It is a sel-proposed problem. G = (,, 3, 4, 5 ) and G = (,, 3, 4 ) = ( x = x x = x x = = ( x.5 ) 3 + ( x x x ) x + 4x.5 3 3) ( x x + x 4 x 3 3) 3 x, x, x3, x4 with the constraint: 0 5

26 Each variable is encoded in 5 bits. 64 uniormly spaced Pareto-optimal solutions are ound or the evaluation o ϒ and σ ϒ. The population size o NSGA-II and the size o archive in PAES are set at 000, and the external population in SPEA is set at 00. As to the legends o notations used in the ollowing tables, please reer to Section Table 9 Perormance o SPEA restarting/inheritance on objective decrement (problem 3) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A Inherit e restart N/A Inherit e e Table 0 Perormance o PAES restarting/inheritance on objective decrement (problem 3) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A e-7 Inherit e e e-6 restart N/A e-7 Inherit e e e-7 Table Perormance o NSGA-II(R) restarting/inheritance on objective decrement (problem 3) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A e-6 Inherit e e e-5 restart N/A e-6 Inherit e e e-6 6

27 Table Perormance o NSGA-II(B) restarting/inheritance on objective decrement (problem 3) t o (s) t d (s) L ϒ σ ϒ η σ d restart N/A e-5 Inherit e e e-7 restart N/A e-6 Inherit e e e-6 Discussions on the results are given in section Remark on Experimental Results The results show that:. When objective decrement happens, during the same time span, the number o solutions ound by MOGAs with inheritance is always more than or equal to those without it. Generally the more evolution is done beore objective decrement, the more solutions will be ound ater objective decrement.. When objective decrement happens, during the same time span, MOGAs with inheritance always converge closer, and more uniormly except or NSGA-II(B). In problem and problem, σ ϒ in NSGA-II(B) with inheritance is larger, but its perormance in the other aspects is much better. The results also showed that generally the more evolution is done beore objective decrement, the closer and more uniormly the solutions will approach the Pareto-ront. 3. As to the coverage o solutions, generally, the metric η given by MOGAs with inheritance in each is better than that o MOGAs without inheritance. Especially 7

28 or NSGA-II with inheritance, its perormance is constantly much better than that o NSGA-II without inheritance. 4. As to the perormance regarding the distribution o solutions, MOGAs with inheritance given by each is better than MOGAs without inheritance most o the time, but slightly worse occasionally. So, when objective decrement occurs, given the same time span, MOGAs with inheritance will ind more solutions closer to the Pareto-ront in a more uniorm way, and the perormance in other aspects will be roughly the same. Besides, the more evolution is done beore objective decrement, the greater advantage the inheritance strategy will have on average. 6. Discussions This part discusses the applicability o inheritance strategy. The experimental results above show that the inheritance strategy can help improve the eiciency o Pareto search in the problems tested. Yet a question ollows: is this strategy applicable to any objective decrement problem? Our answer is that it may not be suitable in some cases. Theorem 3 shows that P P ', but it does not give the percentage o P over P '. In act the percentage may vary rom 0 to. The ollowing problems are two extreme cases: Problem : G =,, ) and G =, ) ( 3 ( 8

29 3 = x = x = sin x with the constraint: x For this problem, P ={ (, ) = x, = x, x [,0] }, and P ' ={ (, ) = x, = x, x [,0] }. So, or any sampling resolution, the percentage o P over P ' is. Problem : G =, ) and G = ) ( = x = x ( with the constraint: x 0 For this problem, P ={ ) }, and P ' ={ ) [,0 ] ( = (. So the percentage o P over P ' is quite small. I the whole region to search is sampled with a higher resolution, the percentage o P over P ' will be very close to 0. Generally, the higher the percentage o P over P ', the more outstanding the advantage o inheritance will be. When the percentage o P over P ' is very small, inheritance may not be helpul, because it may cost too much time to pick out the solutions having good perormance under G when compared to a restarted search. 9

30 A question arises ater the discussion above. How to estimate the percentage o P over P ' and predict the applicability o inheritance strategy or a given problem? This problem is let or uture study. Empirically, we reckon that the inheritance strategy is not applicable to a problem: where the number o objectives deleted is more than. or where the problem ater objective decrement comes down to a singleobjective one. The heuristics suggested above come rom the ollowing observations. Generally, the Pareto set will shrink a lot i more than two objectives are deleted, so the percentage o P over P ' will be quite small. Similarly, i objective decrement turns an MOP into a single-objective problem, due to the removal o competition among objectives, the answer to the problem is no longer a amily o solutions (which is reerred to as Pareto set) but one optimal solution, so the percentage o P over P ' tends to be very small. So, inheritance is unsuitable or these two cases. 7. Conclusions This paper irst analyzed the eect o objective increment on multi-objective optimization. Three theorems have been proved, which state that ater objective increment, unique P-solutions will remain unique Pareto-optimal, at least one member in a non-unique P-group will remain Pareto-optimal, and the Pareto ront beore objective increment is a subset o the Pareto ront ater objective increment with the elements corresponding to the objectives added being truncated. Then the eect o objective 30

31 decrement, which is the reverse operation o objective increment, was discussed. The Pareto ront ater objective decrement is a subset o the Pareto ront beore objective decrement with the elements corresponding to the objectives deleted being truncated. So it makes sense to inherit the population beore objective decrement in the ollowing evolutions to achieve better perormance. Based on this observation, this paper presented the inheritance strategy, which selects well-perorming chromosomes rom the solutions ound beore objective decrement, and reuses them in the ollowing evolutions based on the decremented objective set. Experimental results showed that this strategy can help dierent MOGAs, namely NSGA-II, PAES and SPEA, to be more responsive to the event o objective decrement. More solutions with better quality can be ound during the same time span. 3

32 Appendix: An Example with the Property S > S + on Objective Increment π = sin( x) = x mod3 = 3 x with the constraint: 0 x 30 Assume the original objective set is, ), then S is { x = 0,6,,8,4,30}, and ( S =6. Ater objective 3 is added, S + becomes { x = 0}, and S + =. S > S +. The reason o a shrinking size is that the all the six members in S are non-unique P-solutions, ive o which are dominated by the solution x =0 ater objective increment, and no nonoptimal solutions beore objective increment are upgraded as Pareto-optimal due to objective increment. Yet we can see that the number o Pareto-optimal outputs does not increase. For, ), P =, and P = {(, ) (0,0)}. Ater 3 is added, P + is still equal to, ( = and the only member is P + = {(,, 3 ) (0,0,0)}. So = ' P + = {(, ) = (0,0)}. We can see that ' P P +. Theorem 3 still holds. 3

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